(*
  DaoMath Completeness Theorems
  完备性定理
*)

Require Import Reals.
Require Import Number.
Open Scope R_scope.

(** * 实数完备性 *)

(** Dedekind 分割 *)
Record DedekindCut : Type := mkCut {
  lower_set : R -> Prop;
  upper_set : R -> Prop;
  
  lower_nonempty : exists x, lower_set x;
  upper_nonempty : exists x, upper_set x;
  
  lower_closed : forall x y, lower_set x -> y < x -> lower_set y;
  upper_closed : forall x y, upper_set x -> y > x -> upper_set y;
  
  separation : forall x y, lower_set x -> upper_set y -> x < y;
}.

(** 实数完备性公理 *)
Axiom real_completeness : forall (cut : DedekindCut),
  exists r : R,
  (forall x, lower_set cut x -> x <= r) /\
  (forall x, upper_set cut x -> x >= r).

(** * Cauchy 序列收敛 *)

Definition is_cauchy (u : nat -> R) : Prop :=
  forall eps : R, eps > 0 ->
  exists N : nat, forall n m : nat,
  (n >= N)%nat -> (m >= N)%nat ->
  Rabs (u n - u m) < eps.

Axiom cauchy_criterion : forall (u : nat -> R),
  is_cauchy u ->
  exists l : R, forall eps : R, eps > 0 ->
  exists N : nat, forall n : nat,
  (n >= N)%nat -> Rabs (u n - l) < eps.

(** * 量子 Hilbert 空间完备性 *)

Definition is_cauchy_quantum (u : nat -> QuantumState) : Prop :=
  forall eps : R, eps > 0 ->
  exists N : nat, forall n m : nat,
  (n >= N)%nat -> (m >= N)%nat ->
  real_part (inner_product (u n) (u m)) > 1 - eps.

Axiom hilbert_completeness : forall (u : nat -> QuantumState),
  is_cauchy_quantum u ->
  exists psi : QuantumState, forall eps : R, eps > 0 ->
  exists N : nat, forall n : nat,
  (n >= N)%nat ->
  real_part (inner_product (u n) psi) > 1 - eps.

(** * 拓扑完备性 *)

(** 紧致空间的序列紧性 *)
Axiom compact_sequential : forall (X : Type) (u : nat -> X),
  (* 如果 X 紧致 *)
  exists (v : nat -> X) (phi : nat -> nat),
  (* 则存在收敛子序列 *)
  (forall n, (phi n < phi (S n))%nat) /\
  (forall n, v n = u (phi n)).

